%I A005043 M2587
%S A005043 1,0,1,1,3,6,15,36,91,232,603,1585,4213,11298,30537,83097,227475,625992,
%T A005043 1730787,4805595,13393689,37458330,105089229,295673994,834086421,
%U A005043 2358641376,6684761125,18985057351,54022715451,154000562758
%N A005043 Motzkin sums: a(n) = (n-1)*(2*a(n-1)+3*a(n-2))/(n+1). Also called Riordan 
               numbers or ring numbers.
%C A005043 I'm not sure "Motzkin sums" is a good name for this. It has the proprty 
               that Motzkin(n) = A001006(n) = a(n) + a(n+1), e.g. A001006(4) = 9 
               = 3 + 6 = a(4) + a(5).
%C A005043 Number of 'Catalan partitions', that is partitions of a set 1,2,3,...,
               n into parts that are not singletons and whose convex hulls are disjoint 
               when the points are arranged on a circle (so when the parts are all 
               pairs we get Catalan numbers) - Aart Blokhuis (aartb(AT)win.tue.nl), 
               Jul 04, 2000.
%C A005043 Number of ordered trees with n edges and no vertices of outdegree 1. 
               For n > 1, number of dissections of a convex polygon by nonintersecting 
               diagonals with a total number of n+1 edges. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Mar 06 2002
%C A005043 Number of Motzkin paths of length n with no horizontal steps at level 
               0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 09 2003
%C A005043 Number of Dyck paths of semilength n with no peaks at odd level. Example: 
               a(4)=3 because we have UUUUDDDD, UUDDUUDD and UUDUDUDD, where U=(1,
               1), D=(1,-1). Number of Dyck paths of semilength n with no ascents 
               of length 1 (an ascent in a Dyck path is a maximal string of up steps). 
               Example: a(4)=3 because we have UUUUDDDD, UUDDUUDD and UUDUUDDD. 
               - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2003
%C A005043 Arises in Schubert calculus as follows. Let P = complex projective space 
               of dimension n+1. Take n projective subspaces of codimension 3 in 
               P in general position. Then a(n) is the number of lines of P intersecting 
               all these subspaces. - F. Hirzebruch, Feb 09, 2004
%C A005043 Difference between central trinomial coefficient and its predecessor. 
               Example: a(6)=15=141-126 and (1+x+x^2)^6=...+ 126*x^5 + 141*x^6 +... 
               (Catalan number A000108(n) is difference between central binomial 
               coefficient and its predecessor.) - David Callan (callan(AT)stat.wisc.edu), 
               Feb 07 2004
%C A005043 First diagonal of triangular array in A059346.
%C A005043 a(n) = number of 321-avoiding permutations on [n] in which each left-to-right 
               maximum is a descent (i.e. is followed by a smaller number). For 
               example, a(4) counts 4123, 3142, 2143. - David Callan (callan(AT)stat.wisc.edu), 
               Jul 20 2005
%C A005043 The Hankel transform of this sequence give A000012 = [1, 1, 1, 1, 1, 
               1, 1, . . .]; example : Det([1, 0, 1, 1; 0, 1, 1, 3; 1, 1, 3, 6; 
               1, 3, 6, 15]) = 1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               May 28 2005
%C A005043 The number of projective invariants of degree 2 for n labeled points 
               on the projective line. - Benjamin J. Howard (bhoward(AT)ima.umn.edu), 
               Nov 24 2006
%C A005043 Define a random variable X=trA^2, where A is a 2 X 2 unitary symplectic 
               matrix chosen from USp(2) with Haar measure. The n-th central moment 
               of X is E[(X+1)^n] = a(n). - Andrew V. Sutherland (drew(AT)math.mit.edu), 
               Dec 02 2007
%C A005043 Contribution from Samson Black (sblack1(AT)uoregon.edu), Aug 27 2008: 
               (Start)
%C A005043 Let V be the adjoint representation of the complex Lie algebra sl(2). 
               The dimension of the invariant subspace of the n-th tensor power 
               of V is a(n).
%C A005043 (End)
%C A005043 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 08 2009: 
               (Start)
%C A005043 Starting with offset 3 = iterates of M * [1,1,1,...], where M = a tridiagonal
%C A005043 matrix with [0,1,1,1,...] in the main diagonal and [1,1,1,...] in the 
               super and subdiagonals. (End)
%C A005043 Binomial transform of A126930. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 26 2009]
%D A005043 M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 
               2544-2563.
%D A005043 Almkvist, Gert; Dicks, Warren; Formanek, Edward; Hilbert series of fixed 
               free algebras and noncommutative classical invariant theory. J. Algebra 
               93 (1985), no. 1, 189-214.
%D A005043 D. L. Andrews and T. Thirunamachandran, On three-dimensional rotational 
               averages, J. Chem. Phys., 67 (1977), 5026-5033.
%D A005043 F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 
               (1999) 73-112.
%D A005043 R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series 
               A, 23 (1977), 291-301.
%D A005043 P. Hanlon, Counting interval graphs. Trans. Amer. Math. Soc. 272 (1982), 
               no. 2, 383-426.
%D A005043 B. Howard, J. Millson, A. Snowden and R. Vakil, http://arXiv.org/abs/
               math.AG/0505096, "The projective invariants of ordered points on 
               the line", preprint, 2005.
%D A005043 D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative 
               characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 
               301-320.
%D A005043 J. Riordan, Enumeration of plane trees by branches and endpoints, J. 
               Combin. Theory, A 23 (1975), 214-222.
%D A005043 E. Royer, Interpretation combinatoire des moments negatifs des valeurs 
               de fonctions L au bord de la bande critique, Ann. Sci. Ecole Norm. 
               Sup. (4) 36 (2003), no. 4, 601-620.
%D A005043 H. C. H. Schubert, Math. Annalen, 1895.
%D A005043 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005043 D.-N. Verma (dnverma(AT)math.tifr.res.in), Towards Classifying Finite 
               Point-Set Configurations, preprint, 1997.
%D A005043 F. Yano and H. Yoshida, Some set partition statistics in non-crossing 
               partitions and generating functions, Discr. Math., 307 (2007), 3147-3160.
%H A005043 T. D. Noe, <a href="b005043.txt">Table of n, a(n) for n=0..200</a>
%H A005043 David Callan, <a href="http://www.stat.wisc.edu/~callan/papersother/">
               Riordan numbers are differences of trinomial coefficients</a>.
%H A005043 W. Y. C. Chen, E. Y. P. Deng and L. L. M. Yang, <a href="http://arXiv.org/
               abs/math.CO/0602298">Riordan paths and derangements</a>
%H A005043 E. Deutsch and B. E. Sagan, <a href="http://arxiv.org/pdf/math.CO/0407326">
               Congruences for Catalan and Motzkin numbers and related sequences</
               a>, J. Num. Theory 117 (2006), 191-215.
%H A005043 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=423">
               Encyclopedia of Combinatorial Structures 423</a>
%H A005043 Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/
               abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</
               a>, 2008.
%H A005043 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 
               4 (2001), #01.1.5.
%H A005043 D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, <a href="http:/
               /www.dsi.unifi.it/~merlini/GenRA.ps">On some alternative characterizations 
               of Riordan arrays</a>, Canad. J. Math., 49 (1997), 301-320.
%H A005043 E. Royer, <a href="http://www.carva.org/emmanuel.royer">Interpretation 
               combinatoire des moments negatifs des valeurs de fonctions L au bord 
               de la bande critique</a>
%H A005043 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               IsotropicTensor.html">Link to a section of The World of Mathematics.</
               a>
%F A005043 a(n)=sum((-1)^(n-k)*binomial(n, k)*A000108(k), k=0..n). a(n)=sum(binomial(n+1, 
               k)*binomial(n-k-1, k-1), k=0..ceil(n/2))/(n+1); (for n>1) - Len Smiley 
               (smiley(AT)math.uaa.alaska.edu)
%F A005043 G.f.: (1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)).
%F A005043 G.f.: 2/(1+x+sqrt(1-2*x-3*x^2)) - Paul Peart (ppeart(AT)fac.howard.edu), 
               May 27, 2000
%F A005043 a(n+1) + (-1)^n = a(0)*a(n) + a(1)*a(n-1) + ... + a(n)*a(0) - Bernhart.
%F A005043 a(n) = (1/(n+1)) * Sum_{i} (-1)^i*binomial(n+1, i)*binomial(2*n-2*i, 
               n-i) - Bernhart.
%F A005043 G.f. A(x) satisfies A = 1/(1+x) + x*A^2.
%F A005043 E.g.f.: exp(x)*(BesselI(0, 2*x)-BesselI(1, 2*x)). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Apr 28 2003
%F A005043 a(n)=A001006(n-1)-a(n-1).
%F A005043 a(n+1) = Sum[k=0..n, (-1)^k*A026300(n, k) ], where A026300 is the Motzkin 
               triangle.
%F A005043 a(n)=sum{k=0..n, (-1)^k*binomial(n, k)*binomial(k, floor(k/2))} - Paul 
               Barry (pbarry(AT)wit.ie), Jan 27 2005
%F A005043 a(n) = Sum_{k>=0} A086810(n-k, k) . - Philippe DELEHAM, May 30 2005
%F A005043 a(n+2) = Sum_{ k>=0} A064189(n-k, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               May 31 2005
%F A005043 Moment representation: a(n)=(1/(2*pi))*Int(x^n*sqrt((1+x)(3-x))/(1+x),
               x,-1,3); - Paul Barry (pbarry(AT)wit.ie), Jul 09 2006
%F A005043 Inverse binomial transform of A000108 (Catalan numbers) . - Philippe 
               DELEHAM (kolotoko(AT)wanadoo.fr), Oct 20 2006
%F A005043 a(n) = (2/pi)* Integral_{t_0..pi} (4cos^2(x)-1)^n*sin^2(x)dx - Andrew 
               V. Sutherland (drew(AT)math.mit.edu), Dec 02 2007
%F A005043 G.f.: 1/(1-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-.....(continued fraction). 
               [From Paul Barry (pbarry(AT)wit.ie), Jan 22 2009]
%F A005043 G.f.: 1/(1+x-x/(1-x/(1+x-x/(1-x/(1+x-x/(1-... (continued fraction). [From 
               Paul Barry (pbarry(AT)wit.ie), May 16 2009]
%e A005043 a(5)=6 because the only dissections of a polygon with a total number 
               of 6 edges are: five pentagons with one of the five diagonals and 
               the hexagon with no diagonals.
%p A005043 A005043 := proc(n) option remember; if n <= 1 then 1-n else (n-1)*(2*A005043(n-1)+3*A005043(n-2))/
               (n+1); fi; end;
%p A005043 Order := 20: solve(series((x-x^2)/(1-x+x^2),x)=y,x); #outputs g.f.
%t A005043 a[0] = 1; a[1] = 0; a[n_] := a[n] = (n - 1)*(2*a[n - 1] + 3*a[n - 2])/
               (n + 1); Table[ a[n], {n, 0, 30}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), 
               Jun 14 2005)
%o A005043 (PARI) a(n)=if(n<0, 0, n++; polcoeff(serreverse((x-x^3)/(1+x^3)+x*O(x^n)), 
               n)) /* Michael Somos May 31 2005 */
%Y A005043 Row sums of triangle A020474, first differences of A082395.
%Y A005043 Cf. A126120.
%Y A005043 Sequence in context: A017924 A052827 A033192 this_sequence A099323 A058534 
               A063778
%Y A005043 Adjacent sequences: A005040 A005041 A005042 this_sequence A005044 A005045 
               A005046
%K A005043 nonn,easy,nice,new
%O A005043 0,5
%A A005043 N. J. A. Sloane (njas(AT)research.att.com).
%E A005043 Thanks to Laura L. M. Yang (yanglm(AT)hotmail.com) for a correction, 
               Aug 29, 2004